Electrical calculator



RECORDER INVENTOR. Andria ,Nui'l E1 23 2 a1 PHASE DIFFERENCE 2 Sheets-Sheet l 00 o l.l.OVn

ATTORNEY.

A. N. GLEYZAL ELECTRICAL CALCULATOR May 19, 1953 F'lled June 3, 1949 y 9, 1953 A. N. GLEYZAL 2,639,089

ELECTRICAL CALCULATOR Filed June 3, 1949 2 Sheets-Sheet 2 AMPLIFIER =v 2o -Z=IMPEDANOE ELEMENT 7 ZI=ELEGTRONIG IMPEDANCE ELEMENT 2P 24 N I m+| n ......z l2. 2 z

A! A2 I? m INVEN TOR. I

ATTORNEY.

'. other impedance element;

Patented May 19, 1953 PA g r masseuse Meme.

de Qty" po s'i ble wlth circuits "shown this'disflosmre. i "Afiimportant object of the invention is toprovide a calculating system which is almost instantaneous in operation and which requires only a minimum of computations to determine the values of the electric'afelements of the system ;--for any given se't ofl'eq'uations. Anoth'riobject oft-he invention isltne pro vision of a calculating system which can" "i'sily be set up and operated using standard equib'fnent and ordinary skill.

A further objectof the invention is to provide an electronic device which forms an impedance fe'lementhavingan impedance? which is .ian'yfijre- Idetermmdpositive or.negative..mu1ttiplegof an- ,Other' objectsfiandnmany oLtheattendant advantages, oil-this.ihventionjwill; be, readilyuappi'ieciated has the. same becomes Lhtter understood by ,reierence' to ,the lfiollowing vdetailed ,rdescrip- ...ticnydienuconsidered in connection .withthe .ac

companyingLdraWingsWWheIein:

-"ment, of im i vention.

'1 Fian ee-i aerammat -vv ewwo w imu 15 Vi.V2,

20 meeting each (of s id imictinn points? Yn Vz";

25 ,agesource;'and:impedan lelements .Z1, Z2

area 9 32 1 Z9 f orming a typical part of the electrical calculating network shown in Figure 11.

Figure 3 is a diagrammatic View of an electrical calculati'nfi lcircuit forming a particular 5 embodiment ofthe inventionshown in Figure 1.

Figure 4 is a diagrammatic view of a novel electronic impedance element which may form a part of the electrical calculating network.

Figure 5 is a giagrammatic view showing an 10 additional form of electrical calculating network.

In Figure 1""cf the drawings wherein similar vreferencecharacters denote corrspondi'ng'parts .throughoutitheviews;I0 designates theelec cal network including acfirstset of junction 11V coneifbr each variable} of jmlinear algebraic equations;- ,,a second set "of, junction iipoints El, E211 .E ionegiol' eachoflthe equan ii'mpeidance elements-Z11, .,;,Zme;Qn e lforleach ccefficient .Of the vequation n reetl QDJ Vn with .each Junctionpcint 1,132, J8- sour in man er im e an el ments (:1, i2. -,rnirdirectly.cqm ecti stiun tip o nt v1, v2, '.,'vn;w11;n g llifilminalAg i lleX93 zna directly connectin each J nction? in ini $1,122,; Em with the.oth r,,ter.mina .1 o t e rvoltagec cunce., qe nnfi teiiwheti tn lt B".andlanmnesqi.t teii nctioncpo tsi-ilhnllz, ivn, asgby,a..switcn.. l2 Aways 13.. fie ectivelyic nneeted on mint-Mend ,Q h on 9 1t .c

:YL-YQI w v' 'f a 1. a sui able ehas sdiff e qain i at gi cinr calnof t1, lmneda cevz a a1 0 ,the negative o 1 ymaof lit e rec ergaa the impedances 111,3 2, z gsignil arly tlge reciprocal of the imgedance Z2 is chosen eoual to the negative of the of the'reciprocals of the t e negat H Zm1. seaside mac oxampl the; Figure 2, and let V1, V2 Vn be the voltages respectively betweenpoints V1, V2, Vn and the ground terminalfB, let'Ei be the voltage bete im aQwe-i pedances (1,

It then follows that As the impedances of the impedance elements {1, {2, fn are varied, the resultant voltages V1, V2, V11 relative to ground at junction points V1,V2, .,V11 respectively constitute solu- Circuits such as the one shown in Figure 2 may be set up, one for each equation and the points V1, V2, Vn respectively connected. The voltages V1, V11 constituting solutions may be read by a suitable instrument at the junction points V1, Vn. Figure 1 shows an electrical network for solving the general case of m linear homogeneous algebraic equations, in n variables. For a set of nonhomogeneous linear algebraic equations, the im- 5'11 of the impedance elements {1, {11 or the output of the voltage source may be adjusted so that V11 equals unity.

Where a sinusoidal source of voltage is used, capacitances, inductances, resistances or other suitable electrical devices may be used either singly or in combination as impedance elements. The impedance of such an impedance element is a real or complex number which varies with the frequency and so may be used in a network to correspond to a real or complex coeificient depending on a parameter p in a set of equations provided the parameter is related to the frequency. Positive real coefiicients may conveniently be represented by capacitances since l/C' is a positive quantity in the equation:

where Z represents the charge impedance of a capacitance element and C is the capacity of the element. Negative real coefficients may conveniently be represented by inductances since -1/Yw is a negative quantity in the equation:

where Z represents the charge impedance of an inductance element, L is the inductance of the element and w is the frequency of the alternating voltage source. The charge impedance is the electrical analog impedance of the impedance presented in the mechanical displacement of an object.

The construction of a network for solving a specific set of three linear homogeneous algebraic equations in four variables will now be described.

As an example the following equations are selected:

These equations are a special case of Equations (1), (m) solved by the network shown in Figure 1 where the number n of variables is four and the number m of equations is three. Consequently, an electrical network such as shown in Figure 3 is required. This network is a special case of the network shown in Figure 1 containing four junction points V1, V4 and three junction points E1, E3. Comparing Equations (1), (m) with Equations (1A), (3A), the coeflicient 1/Z11 of V1 in Equation (1) is identified with the coeflicient .0101 of V1 in Equation (1A). Thus Similarly the coefficient 1/Z1z of V2 in Equation (1) is identified with the coefilcient .0700 of V2 in Equation (1A). Thus Likewise the remaining coefiicients appearing in Equations (1A), (3A) may be identified with coefiicients appearing in Equations (1), (m) when n equals four and m equals three. In the present example, a source I IA of alternating voltage having a frequency of 1000 C. P. S. is chosen.

Since all the coeflicients in Equations (1A), (2A), (3A), are positive, it is convenient to use capacitances as impedance elements between points V1, V4 and points E1, E3 in Figure 3. In the example shown, these capacitances are formed by condensers C11, C14; C21, C24; and C31, C34.

In order that the sum of the reciprocals of the impedances about each of the junction points E1, E2, E: be zero, in conformity with Equation (X), it will be necessary to choose negative impedances between points E1, E2, Es respectively and ground. For these balancing impedances it is convenient to use inductance elements L1, La, In as shown in Figure 3.

The impedance of the capacitance element C11 is which has been previously identified with the coeflicient .0101 multiplying V1 appearing in Equation (1A) Thus It is convenient to choose microfarads as units of capacity and therefore in accordance with this choice the condenser C11 has a capacity of .0101 mi. Similarly the capacity of condenser C12 corresponds to the coefiicient multiplying V: in Equation (1A). Thus condenser C12 has a capacity of .0700 mi. Likewise, the capacities in microfarads of the remaining condensers as shown in Figure 3 correspond each with one of the remaining coefficients appearing in Equations (1A), (2A), (3A).

The computation of the balancing impedance elements in accordance with Equation (1) 5. nowheidescribed. .Taking-asan example E1 aszthe junction about Which'theimpedances are. to be:balanced;the,v following; equationmust be satisfied:

" ig-1 11 i i2+ 1a+ 14:

may be computed soas to balance the impedances about junction points Ejz, E3 respectively. The

values thus computed, in henries, are

The network shown in Figure 3 conforms to the 1' foregoing computations and represents the network constructed to solve Equations (1A), (2A), (3A). Upon energizing the network with an alternating voltage. of 1000 C. P. 3., voltage readings taken at points V1, V2, V3, V4 constitute a solution. of Equations (IA), (2A), (3A). solutionwill vary proportionately with the amplitude of th alternating voltage supplied by the source HA. If this amplitude is so adjusted that the voltage at V4. is unity/the voltages appearing at points V1, V2,]Va. constitute the solution of the non-homogeneous equations corresponding to Equations (1A), (2A (3A) provided the proper signs ofthe variables V1, V2, V3 are ascertained. If the voltage .at point V1 is in phase with that at point V4,, the sign of the variable V1 is the same as that of the variable V4.

Tov determine the phase of the voltage at point V1 with respect to that. at point V4, a voltmeter is inserted ther'ebetween. If the sum-of the voltages at point'V1 and atpoint V4 equals the voltage between points V1 and V4, then the voltage at'V1 is out of phase with the. voltage at-Va. If the difference of the voltages at point V1 and at point V4. is equal to the-voltage between points V1 and V4, then the voltage at point V1 is in phase with that at point V4.

'The following table shows the experimental values of V1, V2, V3, V4 obtainedwith the network of Figure 3. and gives. the computed values obtained by solvingEquations (1A), (2A), (3A).

I IExperi.- Percentagemental Computed .Difierence The element or combination of elements Z. torseillirequencies of a suitable range vof; frequencies.

The electronic impedance. elementZ'; comprises an amplifier '20 having; a gcommon1inputand ground terminal 2l,;an input terminal 22 con.- nected to junction point 23, an output terminal 24 connected to one terminal of the impedance element Z whose other terminal-is connected to the input: terminal 22. Finally an impedance R'=RP/(1a) to cancel the plate impedance R? of the amplifier is connected in series between the terminal 2| and a point 2L in'the line as shown.

If the impedance of the impedance element Z is Z and the amplification factor of the amplifier 2!) is a, the effective impedance of the electronic impedance element Z is K RP where z. is the current flowingv through the electronic impedance element Z and V is the voltage at point 23 minus the voltage at point 2|. ."But the current i is also equal to the current flowing through the impedance elementz'. This current iis It is clearth-at the above relationship/holds regardless of whether or'not. thexterminal B1..=is a ground terminal. Thus it 11:2, andR=RP the impedance of the electronic impedance element Z is the negative of the impedance of the element Z.

In order to balance the impedances* 'for example about the junction poin-t ErofFigure 1,- it is only necessary to insert an electronic 'impedance element Z, of the type shown in- Figure-4, in the line between the selected j-unctionpoint E1 and point G, where the impedance Z of the impedance element- Z is-given by This is the relationship ofimpedances connected in parallel so that the impedance element 'Zfcan always be readilyconstructed.

It will therefore beclear that. bychoosingan amplification factor 11:.2, the .efiectiveimpedance of the electronic impedance .elementfZ may be made equal to the negative of'thearbitrary impedance of the impedance element Z.

Figure 5* shows an additional-electrical, network which may be used for solving linearalgebraic equations. This network issimilar tothat shown inFigure l in that it incl-uclesjunction points V1, V2, V11 one for each variable, junction points E1, E2, Em one'for each equation, and impedances Z11 .Zm, Z21 Z211, Z1111 Zmn connecting the junction points V1, V2, V11 to the junction points E1, E2, Em in every possible manner.

The-voltages at'points E1, E2, Em-are amplified by amplifiers A1, A2, Am having their outputs respectively connected to points V1,--V2, -Vm. Iftheamplificatiomfactor-is" a, then In the limit as the absolute value of the amplification factor goes to so these equations become V1 L- V1 LL Z21 22, 0

V1 n =0 ml mn Hence the circuit of Figure if the absolute value of a. is large represents m homogeneous linear algebraic equations in the n variables V1, V2, Vu where the reciprocals i. Zmn

are the coefiicients of the equations. By varying the voltage input of the voltage sources Gm-j-i, Gn connected to Vm+l, Vn all possible solutions of the three equations may be obtained by measuring with a suitable meter 30 the voltage to ground at junction points V1, V2, Vn, as by a suitable selector switch 3|.

Obviously many modifications and variations of the present invention are possible in the light of the above teachings. It is therefore to be understood that within the scope of the appended claims the invention may be practiced otherwise than as specifically described.

The invention described herein may be manufactured and used by or for the Government of the United States of America for governmental purposes without the payment of any royalties thereon or therefor.

I claim:

1. In a system for solving a linear algebraic equation in n variables, an electrical network including a set of junction points one for each of said variables, another junction point, a set of impedance elements one for each coefiicient of the equation directly connecting each junction point of said set of junction points with said othe junction point, a source of voltage, means connecting at least one of the junction points of said set with one terminal of said voltage source, another impedance element directly connecting said other junction point with the other terminal of said voltage source, the sum of the reciprocals of the impedances connected directly to said other junction point being zero, and means for measuring the voltage between any one of said junction points of said set and said other terminal of the voltage source.

2. In the system according to claim 1, means for determining the phase relationship of the voltages appearing at two junction points of said first-mentioned set.

3. In a system for solving a linear algebraic equation in n variables, an electrical network including a set of junction points one for each of said variables, anothe junction point, a set of impedance elements one for each coefiicient of the equation directly connecting each junction point of said set of junction points with said other junction point, a source of voltage, means connecting at least one of the junction points of said set with one terminal of said voltage source, another impedance element directly connecting said other junction point with the other terminal of said voltage source, the sum of the reciprocals of the impedances connected directly to said other junction point being zero, and means for measuring the voltage between any one of said junction points of said set and said other terminal of the voltage source, said other impedance element having an impedance equal to the impedance of said set of impedance elements connected in parallel between said set of junction points and said other junction point, and an electronic impedance reversing device adapted to reverse the impedance of said other impedance element.

4. In a system for solving m linear algebraic equations in n variables, an electrical network in cluding a first set of n junction points comprising a first subset of m junction points and a second subset of junction points where m n, a second set of m junction points, a set of mn impedances connecting each junction point of said first set to each junction point of said second set, a set of m amplifiers with high input impedance, each amplifier having a first input terminal and a first output terminal, said input and output terminals being directly connected, each amplifier having a second input terminal directly connected to a junction point of said second set of junction points, each amplifier also having a second output terminal directly connected to a junction point of said first subset of junction points, n-m, voltage sources each having a first terminal, and a second terminal said first terminals of said voltage sources being connected to said junction points of said second subset, and said second terminals of said voltage sources being each connected to all of said first input terminals of said amplifiers, and a voltage measuring device selectively connected to any one of said first set of n junction points.

5. In a system for solving a linear algebraic equation in 11 variables, an electrical network including a set of junction points one for each of said variables, another junction point, a set of impedance elements one for each coefiicient of the equation directly connecting each junction point of said set of junction points with said other junction point, a source of voltage, means connecting at least one of the junction points of said set with one terminal of said voltage source, another impedance element having an impedance which is any predetermined positive or negative multiple of any desired impedance, said other impedance element directly connecting said other junction point with the other terminal of said voltage source, the sum of said other impedance and. the reciprocals of the impedances representing the coefficients being zero, and means for measuring the voltage between any one of said junction points of said set and said other terminals of the voltage source.

6. In a system for solving a plurality of algebraic equations containing 12. variables, an electrical network including a first set of junction points having a junction for each of said variables, a second set of junction points having a junction for each equation to be solved, a plurality of impedance elements wherein there is one for each coefiicient of the equations, said impedance elements acting to connect each of the first set of junction points with all of the second set of junction points, a source of voltage, means for successively connecting each of the first set of junction points to one terminal of the voltage source, another set of impedance elements connecting the second set of junction points with 20 10 the other terminal of the voltage source, there being one of said other impedance elements for each of the second junction points, the sum of the reciprocals of the impedances directly connected to each of said second set of junction points being zero, and means for measuring the voltage between each of the first set of junction points and said other terminal of the voltage source. 7

ANDRE NOEL GLEYZAL.

References Cited in the file of this patent Reference Data for Radio Engineers by the Federal Telephone and Radio Corp., printed by the Knickerbocker Printing Corp., New York.

Analysis of Problems in Dynamics by Electronic Circuits, Proceedings of the Institute of Radio Engineers and Waves and Electrons vol. 35, No. 5, May 1947. 

